(When) do you need to get a new ID from your new state if you don't drive?

Are there any legal requirements (whether at the state or federal level) regarding whether/when you must get an updated ID

Depends on the state, e.g. yes in Washington state, from RCW 46.20.021:

New Washington residents must obtain a valid Washington driver's license within thirty days from the date they become residents.

OP, this is a frame challenge, or at least a criticism of the question and its title as currently phrased. As phrased: if you don't want to drive a motor vehicle, you don't need to get a new ID from your state, so the answer is "You don't."

In reality, the question assumes sub silentio that a US state (or the federal government) requires a US resident or citizen to obtain an ID. This assumption is not correct.

There is no state-level or federal-government level requirement that people within the borders of the US have or carry an ID. Neither the federal nor the state governments issue a mandatory ID.

All of this is different from many other countries, which do have such a requirement and do issue mandatory national or state IDs for (at least some) people within the country's borders.

In specific cases, however, many people are required to obtain or even carry a governmental ID. States (and I assume the District of Columbia as well) do issue non-drivers'-license ID cards, but no one is required to have or carry such an ID. To be sure, this set of people is smaller than the general set, but it's still a lot of people. For example (and not by way of limitation):

• Drivers of motor vehicles on public streets must have a state-issued (or federal-issued, for the District of Columbia) driver's license from that jurisdiction, and must carry that license when driving a vehicle on a public roadway. There are some exceptions in some places (e.g., very small mopeds, people from other countries or states who are recently-arrived), but have-and-carry-a-license is the general rule for people who drive motor vehicles.

• US Lawful Permanent Residents ("Green Card" holders) are required by the Immigration and Nationality Act to carry the Card "at all times."

• Satisfactory ID evidence is required to be held and presented to TSA by those departing via commercial aviation from a US airport.

• Travel documents (usually passports and visas) are required of those deparing via commercial aviation from a US airport for a non-US airport.

• ID information is required by the airline to be submitted to APIS in order to receive clearance from Customs and Border Protection that the individual is not on a "Don't Fly" list and can be issued a boarding pass.

• To check out a book from a public library, one must have a library card. To obtain a library card, the library will require ID.

This list could continue for pages: Want medical care? Look underage and want to buy a beer? Want to open a credit card or a bank account? Want to enroll in public school? et cetera ad infinitum

There are 50 US states with varying legal requirements, but no US law requires a US citizen to carry ID in the United States.

In practice, even if you do not drive, you can’t get away with using your old ID indefinitely; it will expire at some point. If your driver’s license is expired, even a bouncer or bartender likely would send you away. Generally, it is more of a hassle to get a new ID card if your old one expired ages ago, as well.

Many US states (California and Texas, for example) — perhaps all US states — do allow you to apply for an ID that is just for identification and that is separate from a driver’s license.

If you’re the type of person who travels frequently, though, would suggest that you consider a US Passport Card in addition to your US Passport. A US Passport Card can be used for identification as an alternative to an ID issued by a US state inside and outside of the US as well as for travel inside the US.

If you do not plan to drive in the future, an up-to-date US Passport Card could be more convenient for you than worrying about whether or not you have been in a place long enough to merit applying for a new state-specific ID or if your old one with your parents' address still is really valid, etc.